FOM: Friedman on defending specialized subjects

[Stephen G Simpson]

Dear FOM,

The ongoing discussion of ``applied recursion theory'' is interesting
and valuable, and I want to continue it.  However, the discussion is
in danger of becoming too specialized for a wide audience of
f.o.m. professionals such as exists here on the FOM mailing list.  I
have been wondering how to deal with this problem.

Therefore, I was delighted this morning when Harvey Friedman
telephoned me to offer his reactions.  As usual, Harvey presented some
fascinating insights which are of general intellectual interest,
transcending the relatively narrow subject at hand.  I now want to
share these general insights with you.  Harvey intends to post his own
elaboration of these points later, perhaps tomorrow.

Best wishes to all,
-- Steve

1. DEFENDING SPECIALIZED SUBJECTS

Harvey's first remark to me was that the ongoing discussion of
``applied recursion theory'' (not limited to the FOM list) has been
following a typical, general pattern that is often observed in
academic/intellectual life.  Here is part of the pattern that Harvey
sees.

  a. Some brilliant, fundamental insights lead to the creation of a
     new academic discipline, call it subject X.  (One has in mind the
     insights of G"odel, Turing, et al which led to the creation of
     recursion theory.)  The subject enjoys some spectacular initial
     successes which are of great interest to a wide audience.

  b. Subsquent practitioners of subject X invest a huge amount of
     energy in the development of new concepts, structures, research
     directions, methods, etc., with great commitment and dedication.
     By a combination of scholarly achievement and political skill,
     they establish subject X as a strong presence in the academic
     landscape: publications, hiring, promotion, tenure, etc.

  c. Because subject X has now become well established, the concerns
     and methods of subject X are now free to develop independently,
     as an end in themselves.  Eventually these concerns and methods
     become so specialized that nobody except the most dedicated
     specialists are able to make use of them or derive any
     perceived benefit from them.

  d. Scholars in related fields begin to question or criticize subject
     X by saying that it may be overdeveloped, too specialized, no
     longer relevant to the original fundamental concerns that gave
     rise to it, etc.

  e. Practitioners of subject X defend it on the grounds that

     i. subject X is beautiful and important for its own sake, as a
     kind of pure art-form or sport;

     ii. the insights achieved by subject X are so stunning that
     they are bound to have many valuable applications outside the
     relatively narrow confines of subject X itself.

  f. The critics respond by saying that 

     i. subject X is beautiful only to the practitioners;

     ii. the insights achieved by subject X have not actually
     addressed any relevant concerns, beyond the initial successes;

     iii. the more recent development of subject X has moved it
     farther away from the initial successes, thus making it ever less
     likely to yield anything of general interest.

  g. In the best case, the practitioners of subject X may respond to
     the critics by reforming or redirecting or recasting subject X in
     significant new ways, thus giving subject X a new lease on life.
     This cycle of criticism, reform, criticism, reform, ... can
     continue indefinitely, to the great benefit of subject X and
     related academic subjects.

  h. Another possibility is that the practitioners of subject X may be
     unwilling or unable to change directions.  In this case, their
     response to the critics may be of a different nature.

  i. For instance, the practitioners may respond by insisting that,
     even if the *insights* and *results* achieved by subject X have
     not been particularly valuable for outsiders, the *concepts* and
     *methods* of subject X are of such great general interest that it
     would be a shame not to continue developing them, both for their
     own sake and for the sake of future possible applications, whose
     nature cannot be predicted.

  j. The practitioners of subject X may then run into an intellectual
     impasse, because in order to advance their argument, they need to
     define or delimit the set of concepts and methods that they are
     referring to.  If they define them too narrowly, the claim of
     general interest becomes implausible.  If they define them too
     broadly, the claim of subject X to ownership of these methods and
     concepts becomes ridiculous.  The practitioners of subject X may
     need to walk a fine line, and this may prove difficult or
     impossible.

  k. The practitioners of subject X may then get upset, accuse the
     critics of unfairness, attempt to suppress discussion, etc etc.

  l. (to be continued)

It will be interesting to see how the debate on ``applied recursion
theory'' plays out as an instance of this general pattern.

2. LACK OF EXAMPLES

Harvey's second comment to me is more specifically about recursion
theory, namely the ``pure'' study of recursively enumerable sets
and their degrees of unsolvability.  It has to do with the lack of
examples.  This is a point that Harvey has made previously here on 
FOM, but here it is again.

Usually, when mathematicians undertake an intensive investigation of
some specific structure or class of structures, the need for such an
investigation has already been motivated by a set of specific, natural
examples showing the richness and interest of the subject.  For
instance, group theory was motivated by a wealth of examples such as
matrix groups, permutation groups, symmetries of geometrical figures,
etc.  Contrast this with the r.e. sets and degrees that are so much
beloved by recursion theorists.  The only natural examples known to
date are the original ones, i.e. the halting problem and the complete
r.e. degree.  Thus there is really only one example, and that example
is highly atypical of the way the subject has developed.  It is
reasonable to wonder whether this lack of examples may indicate some
sort of defect or imbalance in the subject.

Harvey has some ideas, which I don't fully understand, about how to
view this as an instance of a general pattern in academic/intellectual
life.  Perhaps Harvey will elaborate later.

In the meantime, let me ask the recursion theorists to respond
straightforwardly to the following question:

   What is the *real* reason for your emphasis or preoccupation with
   pure structural questions concerning the lattice of r.e. sets and
   the semilattice of r.e. degrees?

Some reasons that have been given in the past are: the beauty of the
methods, applications to computer science, possible applications to
pure mathematics, etc.  But somehow these reasons do not ring true.  I
somehow feel that these are not the *real* reasons.  I am asking for
the *real* reason.

-- Steve